da38281c |
1 | /* -*-c-*- |
2 | * |
3 | * $Id: mpmul.h,v 1.1 2000/07/01 11:21:39 mdw Exp $ |
4 | * |
5 | * Multiply many small numbers together |
6 | * |
7 | * (c) 2000 Straylight/Edgeware |
8 | */ |
9 | |
10 | /*----- Licensing notice --------------------------------------------------* |
11 | * |
12 | * This file is part of Catacomb. |
13 | * |
14 | * Catacomb is free software; you can redistribute it and/or modify |
15 | * it under the terms of the GNU Library General Public License as |
16 | * published by the Free Software Foundation; either version 2 of the |
17 | * License, or (at your option) any later version. |
18 | * |
19 | * Catacomb is distributed in the hope that it will be useful, |
20 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
21 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
22 | * GNU Library General Public License for more details. |
23 | * |
24 | * You should have received a copy of the GNU Library General Public |
25 | * License along with Catacomb; if not, write to the Free |
26 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
27 | * MA 02111-1307, USA. |
28 | */ |
29 | |
30 | /*----- Revision history --------------------------------------------------* |
31 | * |
32 | * $Log: mpmul.h,v $ |
33 | * Revision 1.1 2000/07/01 11:21:39 mdw |
34 | * New interface for computing products of many (small) integers. |
35 | * |
36 | */ |
37 | |
38 | #ifndef CATACOMB_MPMUL_H |
39 | #define CATACOMB_MPMUL_H |
40 | |
41 | #ifdef __cplusplus |
42 | extern "C" { |
43 | #endif |
44 | |
45 | /*----- Header files ------------------------------------------------------*/ |
46 | |
47 | #ifndef CATACOMB_MP_H |
48 | # include "mp.h" |
49 | #endif |
50 | |
51 | /*----- Magic numbers -----------------------------------------------------*/ |
52 | |
53 | /* --- How the algorithm works --- * |
54 | * |
55 | * Multiplication on large integers is least wasteful when the numbers |
56 | * multiplied are approximately the same size. When a new multiplier is |
57 | * added to the system, we push it onto a stack. Then we `reduce' the stack: |
58 | * while the value on the top of the stack is not shorter than the value |
59 | * below it, replace the top two elements by their product. |
60 | * |
61 | * Let %$b$% be the radix of our multiprecision integers, and let %$Z$% be |
62 | * the maximum number of digits. Then the largest integer we can represent |
63 | * is %$M - 1 = b^Z - 1$%. We could assume that all of the integers we're |
64 | * given are about the same size. This would give us the same upper bound as |
65 | * that derived in `mptext.c'. |
66 | * |
67 | * However, we're in less control over our inputs. In particular, if a |
68 | * sequence of integers with strictly decreasing lengths is input then we're |
69 | * sunk. Suppose that the stack contains, from top to bottom, %$b^i$%, |
70 | * %$b^{i+1}$%, ..., %$b^n$%. The final product will therefore be |
71 | * %$p = b^{(n+i)(n-i+1)/2}$%. We must now find the maximum stack depth |
72 | * %$d = n - i$% such that %$p > M$%. |
73 | * |
74 | * Taking logs of both sides gives that %$(d + 2 i)(d + 1) > 2 Z$%. We can |
75 | * maximize %$d$% by taking %$i = 0$%, which gives that %$d^2 + d > 2 Z$%, so |
76 | * %$d$% must be approximately %$(\sqrt{8 Z + 1} - 1)/2$%, which is |
77 | * uncomfortably large. |
78 | * |
79 | * We compromise by choosing double the `mptext' bound and imposing high- and |
80 | * low-water marks for forced reduction. |
81 | */ |
82 | |
83 | #define MPMUL_DEPTH (2 * (CHAR_BIT * sizeof(size_t) + 10)) |
84 | |
85 | #define HWM (MPMUL_DEPTH - 20) |
86 | #define LWM (MPMUL_DEPTH / 2) |
87 | |
88 | /*----- Data structures ---------------------------------------------------*/ |
89 | |
90 | typedef struct mpmul { |
91 | size_t i; |
92 | mp *v[MPMUL_DEPTH]; |
93 | } mpmul; |
94 | |
95 | #define MPMUL_INIT { 0 } |
96 | |
97 | /*----- Functions provided ------------------------------------------------*/ |
98 | |
99 | /* --- @mpmul_init@ --- * |
100 | * |
101 | * Arguments: @mpmul *b@ = pointer to multiplier context to initialize |
102 | * |
103 | * Returns: --- |
104 | * |
105 | * Use: Initializes a big multiplier context for use. |
106 | */ |
107 | |
108 | extern void mpmul_init(mpmul */*b*/); |
109 | |
110 | /* --- @mpmul_add@ --- * |
111 | * |
112 | * Arguments: @mpmul *b@ = pointer to multiplier context |
113 | * @mp *x@ = the next factor to multiply in |
114 | * |
115 | * Returns: --- |
116 | * |
117 | * Use: Contributes another factor to the mix. It's important that |
118 | * the integer lasts at least as long as the multiplication |
119 | * context; this sort of rules out @mp_build@ integers. |
120 | */ |
121 | |
122 | extern void mpmul_add(mpmul */*b*/, mp */*x*/); |
123 | |
124 | /* --- @mpmul_done@ --- * |
125 | * |
126 | * Arguments: @mpmul *b@ = pointer to big multiplication context |
127 | * |
128 | * Returns: The product of all the numbers contributed. |
129 | * |
130 | * Use: Returns a (large) product of numbers. The context is |
131 | * deallocated. |
132 | */ |
133 | |
134 | extern mp *mpmul_done(mpmul */*b*/); |
135 | |
136 | /* --- @mp_factorial@ --- * |
137 | * |
138 | * Arguments: @unsigned long i@ = number whose factorial should be |
139 | * computed. |
140 | * |
141 | * Returns: The requested factorial. |
142 | */ |
143 | |
144 | extern mp *mp_factorial(unsigned long /*i*/); |
145 | |
146 | /*----- That's all, folks -------------------------------------------------*/ |
147 | |
148 | #ifdef __cplusplus |
149 | } |
150 | #endif |
151 | |
152 | #endif |